Abstract
We propose a new set of partial differential equations which can be seen as a generalization of the classical eikonal and transport equations, to allow for solutions with multiple phases. The traditional geometrical optics pair of equations do not include solutions with multiple phases, corresponding to crossing waves. The new partial differential equations form a hyperbolic system of conservation laws with source terms. They are derived from a closure of the kinetic formulation of geometrical optics. Numerical examples are presented.
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References
Y. Brenier and L. Corrias, Capturing multivalued solutions, CAM Report 94–46, Department of Mathematics, UCLA, 1994.
Y. Brenier and E. Grenier, On the model of pressureless gases with sticky particles, CAM Report 94–45, Department of Mathematics, UCLA, 1994.
M. Crandall and P. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1–42.
W. E, Yu. G. Rykov, and Ya. G. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, to appear.
B. Engquist, E. Fatemi and S. Osher, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation, CAM Report 93–05, Department of Mathematics, UCLA, 1993.
G.-S. Jiang and E. Tadmor, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, CAM Report 96–36, Department of Mathematics, UCLA, 1996.
R. J. LeVeque. Numerical Methods for Conservation Laws, Birkhäuser, 1992.
H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, Journal of Computational Physics, 87 (2) (1990), 408–463.
F. Qin et al, Finite-difference solution of the eikonal equation along expanding wave-fronts, Geophysics, 57 (3) (1992), 478–487.
O. Runborg, Multiphase computations in geometrical optics, Licentiate’s Thesis TRITA-NA-9609, Department of Numerical Analysis and Computing Science, KTH, 1996.
J. van Trier and W. W. Symes, Upwind finite-difference calculation of traveltimes, Geophysics, 56(6) (1991), 812–821,.
J. Vidale, Finite-difference calculation of traveltimes, Bulletin of the Seismological Society of America, 78(6) (1988), 2062–2076.
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Engquist, B., Runborg, O. (1999). Multiphase Computations in Geometrical Optics. In: Fey, M., Jeltsch, R. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8720-5_30
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DOI: https://doi.org/10.1007/978-3-0348-8720-5_30
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